we know that tile reliability of the non-maintained linear consecutive k-out- of-n system .... Simplified reliabilities for consecutive k-out-of-n systems, SIAM J. Alge-.
Ann. Inst. Statist. Math. Vol. 44, No. 4, 605-612 (1992)
CONSECUTIVE k-OUT-OF-n SYSTEMS WITH MAINTENANCE STAVROS G. PAPASTAVRIDIS AND MARKOS V. KOUTRAS Department of Mathematics, University of Athens, Panepistemiopolis, 157 84 Athens, Greece
(Received May 24, 1990; revised May 27, 1991)
A b s t r a c t . A consecutive k-out-of-n system consists of n linearly or cyclically ordered components such that the system fails if and only if at least k consecutive components fail. In this paper we consider a maintained system where each component is repaired independently of the others according to an exponential distribution. Assuming general lifetime distributions for system's components we prove a limit theorem for the time to first failure of both linear and circular systems.
Key words and phrases: Consecutive k-out-of-n systems, reliability bounds, maintenance, time to failure, Weibull limit theorem.
1.
Introduction
Consecutive k-out-of-n systems have been used to model telecommunications, oil pipelines, vacuum systems in accelerators, c o m p u t e r ring networks and space relay stations. A consecutive k-out-of-n system consists of n components arranged on a line or on a circle. Each c o m p o n e n t has two states: up (working) and down (failed). T h e system is considered to be down if and only if at least k consecutive c o m p o n e n t s are down. During the last decade, a lot of research has been done to c o m p u t e various reliability characteristics of consecutive k-out-of-n systems for the case of the non-maintained systems, D e r m a n et al. (1982), Fu (1985, 1986), Hwang (1986), Papastavridis (1987). In the present paper we examine the situation of a maintained consecutive k-out-of-n system where, each c o m p o n e n t is separately maintained and undergoes a perfect repair every time it goes down. Repair starts immediately after a component's breakdown, and the repair time is independent of the other components. In Section 2 we derive lower and u p p e r bounds for the distribution of the time T~ of first failure of a consecutive k-out-of-n linear system with maintenance. B o t h bounds are expressed via the distribution of time of first failure of parallel subsystems. We mention here t h a t a lot of research work is available for the first failure time of parallel systems, Gaver (1964), Branson and Shah (1971), B h a t (1973), Ross and Schechtman (1979), which can be efficiently used for the evaluation of our bounds. In Section 3 we prove t h a t in the case of a consecutive 605
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k-out-of-n linear system with general lifetimes and exponential repair times, the distribution of T~ approaches the Weibull distribution, as n becomes large. The proof is based on the inequalities given in Section 2 and Brown's analysis of parallel systems (1975). Finally, in Section 4 we show t h a t the same limit theorem applies also in the circular system. The corresponding results for the non-maintained case have been proved by Papastavridis (1987). Recently Chao and F~l (1989) proved a limiting theorem of similar nature for a very large class of systems which includes consecutive k-out-of-~ systems as a special case, using a different set of assumptions. 2.
Bounds for the linear case
Consider a system consisting of n separately maintained independent components arranged on a line and numbered 1, 2 , . . . , n. The i-th component is assumed to have a lifetime distribution Fi and repair distribution Gi, i = 1, 2 , . . . , n. At time t = 0 all components are up (working), and thereon, each component alternates between intervals in which it is up and in which it is down. Suppose t h a t the system works according to the "consecutive k-out-of-n" principle (i.e. it is considered down at time t~ if and only if there are at least k consecutive failed components at time t) and denote by T~ the time of the first system failure. For a fixed t > 0 we introduce the next events related to the components' performance in the time interval [0, t]. (a) Ci, i = 1, 2 . . . . . n is the event t h a t the i-th component went down at least once in the time interval [0, t]. Conventionally we set Co = ~. (b) S~. i = 1.2 . . . . . n - k + 1 is the event t h a t there is an instance in the time interval [0, t], when all components i, i + 1 , . . , , i + k - 1 are down. It is obvious that the failure of the maintained consecutive k-out-of-n system in the time interval [0, t], means the occurrence of at least one of the events S~, i = 1,2, . . . . n - k + 1 . Throughout this paper Pr(E) denotes the probability of the event E. If El, E2 are any two events then E1 - E2 will be used for the set theoretic difference among sets, E1E~ for the intersection, E ~ for the complement etc. Before going to the proof of the main result of this section, we will take care of a small detail, that will be needed in the sequel. LEMMA 2.1. If A~, i = k , k + 1 . . . . . n is the event that the consecutive kout-of-i system consistin9 of components 1, 2 . . . . . i stays continuously up in time interval [0, t], then Pr(Ci k+l I Ai) 2k the proof follows immediately from inequality
Pr(A'zC[_k+l) : Pr(A'/_k)Pr(C~ k , l ) -< Pr(A'/)Pr(C~_k+l).
[]
We are now ready to prove the following theorem providing an upper bound for the reliability of a linear consecutive k-out-of-n system with maintenance.
CONSECUTIVE SYSTEMS WITH MAINTENANCE THEOREM 2.1.
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For the time T~ of the first system failure it holds that n-k+l
(2.1)
Vr(Tn _> t) _< H
{1-Pr(Si)+Pr(Si)Pr(Ci-1)}.
i=1
PROOF. Since {Ai}i=k .....~ is a monotone decreasing sequence of events (sets) we have that Pr(AiA~+I) = Pr(Ai) - Pr(Ai+l) and n--1
(2.2)
Pr(T,~ > t) = Pr(An) = Pr(Ak) H
Pr(Ai+l) Pr(Ai)
i=k re-1
= Pr(Ak) H { 1 - Pr(A'i+ 1 [ Ai)}. i=k
Conditioning on the event C~_k+ 1 that the i - k + 1 component is continuously up in the time interval [0, t], and taking Lemma 2.1 into account, we obtain Pr(A'~+ 1 [Ai) > Pr(A~+ 1 ]AiC~_k+l)Pr(C~_k+ 1 [Ai) > Pr(A'i_ 1 I AiC~_k+l){1 - Pr(Ci_k+l)}. But, it is not difficult to observe that Pr(A~+ 1 I AiC[_k+l) = Pr(Si-k+2)
(2.3)
and the proof of the theorem follows immediately. [] The theorem above is an improvement of a result given by Fu ((1986), p. 317, Theorem 1), which is stated there for the non-maintained case. Our proof consists of a sharpening of Fu's ideas. As numerical computations performed by Fu indicate, the previous theorem provides a good approximation of system's reliability for the non-maintained case, and we believe that it is of independent interest. In the next theorem we give two lower bounds for the reliability of a linear consecutive k-out-of-n system with maintenance. The first of them is valid for any life and repair time distributions of the components. The second provides a significantly better approximation to system's reliability on the cost of certain distributional restrictions on components' life and repair times. THEOREM 2.2.
(a) For the time Tr~ of the first system failure it holds that n-k+1
Pr(T~ _> t) _> H
{1 - Pr(Si) - Pr(Ci-1)}.
i:l
(b) If each component has exponential failure distribution and the repair time distributions have decreasing repair rate (i.e. repair distributions are DFR) then n-k+1
Pr(T~_>t)_
H
{1-Pr(Si)}.
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PROOF.
(a) From the total probability theorem we obtain
Pr(A~i+l I Ai) = Pr(A~i+l I AiCi-k+l) Pr(C~-k+l I Ai) + Pr(A~i+l l A i t y - k + 1 ) Pr(C~-k+l I Ai) _< Pr(C~-k+l ] Ai) + Pr(A'~+ 1 I d~C~-k+J) and making use of Lemma 2.1 and (2.3) we conclude that Pr(A~i+l I Ai) 0. Furthermore, let FD denote the cumulative distribution function of the time to first failure of a parallel system with k components, which are separately maintained. The proof of the main result of this section requires the knowledge of the McLaurin's expansion first term of FD. Adjusting Brown's arguments (1975) to our model, we may introduce the following function
P(t) = kA(t)p(t)q k-l(t) which expresses the probability that the system is working at time t and goes down in the time interval [t, t + dt], divided by dt. In other words, P(t) is the derivative of the expected number of failures of the system in [0, t]. The next two lemmas will be crucial in the sequel. LEMMA 3.2.
There is a function M such that
~ot P(s)ds
(3.4)
PROOF.
= FD(t) + (FD * M)(t).
See Brown (1975) p. 368 and p. 392, Appendix 8. []
LEMMA 3.3. If F is of the form F(t) = ()~t)a + o(t a) and the repair times are exponential with mean 1/p, then
FD(t) = (At) "k + o(tak).
(3.5)
PROOF.
The results of Lemma 3.1 allow us to write
P(t) = k a ~ ( ~ t ) a-1 • 1. (Ata) k-1 + o(t ak-1) = kaAakt ak-1 + o ( t a k - 1 ) . Hence
~ot P(s)ds = (At) ak + o(t ak)
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AND
M A R K O S V. K O U T R A S
and (3.5) is now easily deduced from (3.4). [] We conjecture t h a t (3.5) is true for general repair distribution too, but we were unable to prove it. This is the only obstacle in stating the limit theorems of this paper for general repair distribution. We are now ready to prove a limit theorem concerning the time T, to first failure of a linear consecutive k-out-of-n system with iid components having arbitrary lifetime distributions and exponential repair times. THEOREM 3.1. If F is of the form F( t ) = (At) ~ + o( t ~) and the repair times are exponential with mean 1/p., then the random variable nl/~kT, is asymptotically Weibull i.e. lim Pr(nl/akT, _< t) = 1 - exp[-(At)~k]. rt --* O(;
PROOF.
Let t,~ = tn -1/~k. Theorem 2.1 gives Pr(Tn >_ t)
